In this paper, we develop an efficient and energy stable fully-discrete local discontinuous Galerkin (LDG) method for the Cahn–Hilliard–Hele–Shaw (CHHS) system. The semi-discrete energy stability of the LDG method is proved firstly. Due to the strict time step restriction ( t = O( x4)) of explicit time discretization methods for stability, we introduce a semi- implicit time integration scheme which is based on a convex splitting of the discrete Cahn– Hilliard energy. The unconditional energy stability has been proved for this fully-discrete LDG scheme. The fully-discrete equations at the implicit time level are nonlinear. Thus, the nonlinear Full Approximation Scheme (FAS) multigrid method has been applied to solve this system of algebraic equations, which has been shown the nearly optimal complexity numerically. Numerical results are also given to illustrate the accuracy and capability of the LDG method coupled with the multigrid solver.

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We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M, g) that are C0-asymptotic to Schwarzschild of mass m > 0. Refining an argument due to H. Bray, we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all sufficiently large volumes, and that they are close to centered coordinate spheres. This implies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume. This confirms a conjecture of H. Bray. Our results are consistent with the uniqueness results for volumepreserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work.

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold$X$. Given a big (1, 1)-cohomology class$α$on$X$(i.e. a class that can be represented by a strictly positive current) and a positive measure$μ$on$X$of total mass equal to the volume of$α$and putting no mass on pluripolar sets, we show that$μ$can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in$α$. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if$μ$has$L$^{1+$ε$}-density with respect to Lebesgue measure. If$μ$is smooth and positive everywhere, we prove that$T$is smooth on the ample locus of$α$provided$α$is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.

We study the solvability problem for the multidimensional Riccati equation −∇$u$=|∇$u$|^{q}+ω, where$q$>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation −Δ$u$−ω$u$=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions on$R$^{n}in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type$−Lu=f(x, u, ∇u)$+ω where, and$L$is a uniformly elliptic operator.